lecture of k.schwarz on hyperfine interactions
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Hyperfine interactions
Karlheinz SchwarzInstitute of Materials Chemistry
TU Wien
Some slides were provided by Stefaan Cottenier (Gent)
nuclear point chargesinteracting with
electron charge distribution
hyperfine interaction
all aspects of the nucleus-electron interaction
which go beyondan electric point charge for a nucleus.
=
Definition :
electricpoint
charge
electricpoint
charge
• volume• shape
N
S
electricpoint
charge
• volume• shape• magnetic moment
How to measure hyperfine interactions ?
This talk:• Hyperfine physics• How to calculate HFF with WIEN2k
• NMR• NQR• Mössbauer spectroscopy• TDPAC• Laser spectroscopy• LTNO• NMR/ON• PAD• …
• Definitions
• magnetic hyperfine interaction
• electric quadrupole interaction
• isomer shift
• summary
Content
N
S
N
S
N
S
N
S
N
S
N
S
-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
N
S
-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
N
S
-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
N
S
-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
N
S
-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
Classical Quantum(=quantization)
e.g. I =1
m =+1
m =0
m =-1
-1.5
-1
-0.5
0
0.5
1
1.5
0 30 60 90 120 150 180
Ener
gy (u
nits
: µB)
E
Classical Quantum(=quantization)
e.g. I =1
m =+1
m =0
m =-1
Hamiltonian :
nuclear property electron property
N
S
(vector) (vector)
interaction energy (dot product) :
Btot = Bdip + Borb + Bfermi + Blat
Source of magnetic fields at a nuclear site in an atom/solid
Btot = Bdip + Borb + Bfermi + Blat
Bdip = electron as bar magnet
Source of magnetic fields at a nuclear site in an atom/solid
Btot = Bdip + Borb + Bfermi + Blat
Bdip = electron as bar magnet
rv
-e
L
Borb
Morb
I
Borb = electron as current loop
Source of magnetic fields at a nuclear site in an atom/solid
Btot = Bdip + Borb + Bfermi + Blat
Bdip = electron as bar magnet
rv
-e
L
Borb
Morb
I
Borb = electron as current loop
BFermi = electron at nucleus
2s
2p
Source of magnetic fields at the nuclear site in an atom/solid
Btot = Bdip + Borb + Bfermi + Blat
Bdip = electron as bar magnet
rv
-e
L
Borb
Morb
I
Borb = electron as current loop
BFermi = electron at nucleus
2s
2p
Source of magnetic fields at a nuclear site in an atom/solid
Blat = neighbours as bar magnets
Magnetic hyperfine field
In regular scf file::HFFxxx (Fermi contact contribution)
After post-processing with LAPWDM :• orbital hyperfine field (“3 3” in case.indmc)• dipolar hyperfine field (“3 5” in case.indmc)in case.scfdmup
After post-processing with DIPAN :• lattice contributionin case.outputdipan
more info:UG 7.8 (lapwdm)UG 8.3 (dipan)
How to do it in WIEN2k ?
Mössbauer spectroscopy:
Isomer shift: δ = α (ρ0Sample – ρ0
Reference); α=-.291 au3mm s-1
proportional to the electron density ρ at the nucleus
Magnetic Hyperfine fields: Btot=Bcontact + Borb + Bdip
Bcontact = 8π/3 µB [ρup(0) – ρdn(0)] … spin-density at the nucleus
… orbital-moment
… spin-moment
S(r) is reciprocal of the relativistic mass enhancement
Verwey transition in YBaFe2O5
CO structure: Pmma VM structure: Pmmma:b:c=2.09:1:1.96 (20K) a:b:c=1.003:1:1.93 (340K)
Fe2+ and Fe3+ chains along b Fe2.5+
a b
c
Charged orderedFe2+ Fe3+
Valence mixedFe2.5+
Ba
Y
DOS: GGA+U vs. GGAGGA+U GGA
insulator, t2g band splits metallic single lower Hubbard-band in VM splits in CO with Fe3+ states lower than Fe2+
Difference densities ∆ρ=ρcryst-ρatsup
CO phase VM phaseFe2+: d-xzFe3+: d-x2
O1 and O3: polarized toward Fe3+
Fe: d-z2 Fe-Fe interactionO: symmetric
YBaFe2O5 HFF, IS and EFG with GGA+U, LDA/GGA
CO
VM
HFF(Fe2+)
HFF(Fe3+)
HFF(Fe2.5+)
• Definitions
• magnetic hyperfine interaction
• electric quadrupole interaction
• isomer shift
• summary
Content
Electric Hyperfine-Interaction
between nuclear charge distribution (σ) and external potential
Taylor-expansion at the nuclear position∫= dxxVxE n )()(σ
+
∂∂∂
+
∂∂
+
=
∑ ∫ ∫
∫∑
ijji
ji
ii i
dxxxxxx
V
dxxxx
V
ZVE
)()0(21
)()0(
2
0
σ
σ
direction independent constant
electric field xnuclear dipol moment (=0)
electric fieldgradient xnuclear quadrupol moment Q
higher terms neglected
nucleus with charge Z, but not a sphere
• Force on a point charge:
• Force on a point charge:
• Force on a general charge:
-Q
-Q
-Q
-Q
-Q
-Q
-Q
2
-Q
-Q
-4,05
-4,04
-4,03
-4,02
-4,01
-4,00
-3,99
-3,98
-3,970 30 60 90 120 150 180
E-tm0
θ (deg)
Ener
gy (u
nits
e2 /
4πε 0
)
2
θ (deg)
Ener
gy (u
nits
e2 /
4πε 0
)
-4,05
-4,04
-4,03
-4,02
-4,01
-4,00
-3,99
-3,98
-3,970 30 60 90 120 150 180
E-tm0
-4,05
-4,04
-4,03
-4,02
-4,01
-4,00
-3,99
-3,98
-3,970 30 60 90 120 150 180
E-tm0
θ (deg)
Ener
gy (u
nits
e2 /
4πε 0
)
m=±1
m=0
e.g. I = 1
nuclear property electron property(tensor – rank 2 )
interaction energy (dot product) :
(tensor – rank 2 )
Electric field gradients (EFG)
Nuclei with a nuclear quantum number I 1 have an electrical quadrupole moment Q
Nuclear quadrupole interaction (NQI) can aid to determine the distribution of the electronic charge surrounding such a nuclear site
Experiments NMR NQR Mössbauer PAC
heQ /Φ≈νEFG traceless tensorΦ
VV ijijij2
31
∇−=Φ δ
jiij xx
VV∂∂
∂=
)0(2
0=++ zzyyxx VVVwith traceless
⇒
cc
bc
ac
cb
bb
ab
ca
ba
aa
VVV
VVV
VVV
zz
yy
xx
VV
V00
0
0
00 |Vzz| |Vyy| |Vxx|≥ ≥
//////
zz
yyxx
VVV −
=η
EFG Vzzasymmetry parameterprincipal component
Nuclear electronic
≥
First-principles calculation of EFG
, 1192
Previous: point charge model andSternheimer factor to experimental value
Electronic structure of Li3N
• The charge anisotropy around N differs strongly between • muffin-tin and • full-potential
• affecting the EFG.
Nuclear quadrupole moment of 57Fe, 3545
From the slope between the theoreical EFG and experimental quadrupole
splitting ΔQ (mm/s) the nuclear quadrupole
moment Q of the most important Mössbauer nucleus is found to be about twice as large (Q=0.16 b) as so far inliterature (Q=0.082 b)
theoretical EFG calculations:
∫ ∑=′′−
′=
∂∂∂
=LM
LMLMcji
ij rYrVrdrr
rrVxx
VV )ˆ()()()()0(2 ρ
2220
2220
20
21
21
)0(
VVV
VVV
rVV
xx
yy
zz
+−∝
−−∝
=∝
EFG is a tensor of second derivatives of VC at the nucleus:
[ ]
[ ]222 )(211
)(211
)(
3
3
320
zyzxzyxxyd
dzz
zyxp
pzz
dzz
pzzzz
dddddr
V
pppr
V
VVdrr
YrV
−+−+∝
−+∝
+=∝
−
∫ρ
Cartesian LM-repr.
EFG is proportional to differences in orbital occupations
High temperature superconductor
YBa2Cu3O7
Electronic structure Charge density, EFG
EFG (electric field gradient)
K.Schwarz, C.Ambrosch-Draxl, P.Blaha,Phys.Rev. B 42, 2051 (1990)
EFG at O sites in YBa2Cu3O7
Interpretation of the EFG (measured by NQR) at the oxygen sites
px py pz Vaa Vbb Vcc
O(1) 1.18 0.91 1.25 -6.1 18.3 -12.2
O(2) 1.01 1.21 1.18 11.8 -7.0 -4.8
O(3) 1.21 1.00 1.18 -7.0 11.9 -4.9
O(4) 1.18 1.19 0.99 -4.7 -7.0 11.7
ppp
zz rnV ><∆∝ 3
1)(
21
yxzp pppn +−=∆
Asymmetry count EFG (p-contribution)
EFG is proportional to asymmetric charge distributionaround given nucleus
Cu1-d
O1-py
EF
partly occupied
y
x
z
non-bonding O-p,C
O1
EFG (1021 V/m2) in YBa2Cu3O7
Site Vxx Vyy Vzz η Y theory -0.9 2.9 -2.0 0.4 exp. - - - - Ba theory -8.7 -1.0 9.7 0.8 exp. 8.4 0.3 8.7 0.9 Cu(1) theory -5.2 6.6 -1.5 0.6 exp. 7.4 7.5 0.1 1.0 Cu(2) theory 2.6 2.4 -5.0 0.0 standard LDA calculations give exp. 6.2 6.2 12.3 0.0 good EFGs for all sites except Cu(2) O(1) theory -5.7 17.9 -12.2 0.4 exp. 6.1 17.3 12.1 0.3 O(2) theory 12.3 -7.5 -4.8 0.2 exp. 10.5 6.3 4.1 0.2 O(3) theory -7.5 12.5 -5.0 0.2 exp. 6.3 10.2 3.9 0.2 O(4) theory -4.7 -7.1 11.8 0.2 exp. 4.0 7.6 11.6 0.3
K.Schwarz, C.Ambrosch-Draxl, P.Blaha, Phys.Rev. B42, 2051 (1990) D.J.Singh, K.Schwarz, P.Blaha, Phys.Rev. B46, 5849 (1992)
Cu partial charges in YBa2Cu3O7
a transfer of 0.07 e into the dz2 would increase the EFG from -5.0 by
bringing it to -11.6 inclose to the Experimental value (-12.3 1021 V/m2)
How to do it in WIEN2k ?
Electric-field gradient
In regular scf file::EFGxxx:ETAxxxMain directions of the EFG
Full analysis printed in case.output2 if EFG keyword in case.in2 is put (UG 7.6)(split into many different contributions)
more info:• Blaha, Schwarz, Dederichs, PRB 37 (1988) 2792• EFG document in wien2k FAQ (Katrin Koch, SC)
} 5 degrees of freedom
Ba3Al2F12
Rings formed by four octahedra sharing
corners
α-BaCaAlF7
Isolated octahedra
Ca2AlF7, Ba3AlF9-Ib,
α-BaAlF5
Isolated chains of octahedra linked by
corners
α-CaAlF5, β-CaAlF5,
β BaAlF γ
EFGs in fluoroaluminates10 different phases of known structures from CaF2-AlF3,
BaF2-AlF3 binary systems and CaF2-BaF2-AlF3 ternary system
νQ and ηQ calculations using XRD data
Attributions performed with respect to the proportionality between |Vzz|and νQ for the multi-site compounds
Calculated ηQ
0,0 0,2 0,4 0,6 0,8 1,0
Exp
erim
enta
l ηQ
0,0
0,2
0,4
0,6
0,8
1,0
AlF3 α-CaAlF5 β-CaAlF5 Ca2AlF7 α-BaAlF5 β-BaAlF5 γ-BaAlF5 Ba3Al2F12 Ba3AlF9-Ib β-Ba3AlF9
α-BaCaAlF7
RegressionηQ,mes. =ηQ, cal.
Calculated Vzz (V.m-2)
0,0 1,0e+21 2,0e+21 3,0e+21
Exp
erim
enta
l νQ (H
z)
0,0
2,0e+5
4,0e+5
6,0e+5
8,0e+5
1,0e+6
1,2e+6
1,4e+6
1,6e+6
1,8e+6AlF3 α-CaAlF5 β-CaAlF5 Ca2AlF7 α-BaAlF5 β-BaAlF5 γ-BaAlF5 Ba3Al2F12 Ba3AlF9-Ibβ-Ba3AlF9 α-BaCaAlF7 Régression
ηQ,exp = 0,803ηQ,cal R2 =
0,38 νQ = 4,712.10-16 |Vzz| with R2 =
0,77
Important discrepancies when structures are used which were determined from X-ray powder diffraction data
M.Body, et al., J.Phys.Chem. A 2007, 111, 11873 (Univ. LeMans)
Very fine agreement between experimental and calculated values
Calculated Vzz (V.m-2)
0,0 5,0e+20 1,0e+21 1,5e+21 2,0e+21 2,5e+21 3,0e+21
Exp
erim
enta
l νQ (H
z)
0,0
2,0e+5
4,0e+5
6,0e+5
8,0e+5
1,0e+6
1,2e+6
1,4e+6
1,6e+6
1,8e+6
AlF3 α-CaAlF5 β-CaAlF5
Ca2AlF7 α-BaAlF5 β-BaAlF5 γ-BaAlF5 Ba3Al2F12 Ba3AlF9-Ib β-Ba3AlF9 α-BaCaAlF7 Regression
νQ = 5,85.10-16 Vzz R2 = 0,993
Calculated ηQ
0,0 0,2 0,4 0,6 0,8 1,0
Exp
erim
enta
l ηQ
0,0
0,2
0,4
0,6
0,8
1,0
AlF3 α-CaAlF5 β-CaAlF5 Ca2AlF7 α-BaAlF5 β-BaAlF5 γ-BaAlF5 Ba3Al2F12 Ba3AlF9-Ib β-Ba3AlF9
α-BaCaAlF7 RegressionηQ,exp. =ηQ, cal.
ηQ, exp = 0,972 ηQ,cal
R2 = 0,983
νQ and ηQ after structure optimization
NMR shielding, chemical shift:
• Definitions
• magnetic hyperfine interaction
• electric quadrupole interaction
• isomer shift
• summary
Content
Mössbauer spectroscopyIsomer shift: charge transfer too small in LDA/GGAHyperfine fields: Fe2+ has large Borb and BdipYBaFe2O5 HFF, IS and EFG with GGA+U, LDA/GGA
CO
VM
Cu(2) and O(4) EFG as function of r
EFG is determined by the non-spherical charge density inside sphere
Cu(2)
O(4)
∫∫∑ =∝= drrrdrr
YrVYrr zzLM
LMLM )()()()( 20320 ρρρρ
final EFG
r
r r
rr
EFG contributions:
Depending on the atom, the main EFG-contributions come from anisotropies (in occupation or wave function) semicore p-states (eg. Ti 3p much more important than Cu 3p) valence p-states (eg. O 2p or Cu 4p) valence d-states (eg. TM 3d,4d,5d states; in metals “small”) valence f-states (only for “localized” 4f,5f systems)
usually only contributions within the first nodeor within 1 bohr are important.
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